Ensemble canonique | EPFL Graph Search

En physique statistique, l’ensemble (ou situation) canonique est un ensemble statistique introduit par le physicien américain Josiah Willard Gibbs. Il correspond au cas d'un système physique de volume V donné et contenant un nombre fixe N de particules, en interaction avec un autre système, appelé réservoir ou thermostat, beaucoup plus grand que le système considéré et avec lequel il peut échanger de l'énergie mais pas de matière. Le thermostat se comporte comme un réservoir supposé infini d'énergie, la réunion des deux systèmes étant considérée comme isolée. Le couplage entre le système étudié et le réservoir est considéré comme faible, c'est-à-dire que l'état du réservoir n'est pas modifié quels que soient les échanges d'énergie entre lui et le système. Un exemple d'une telle situation peut être donné par une bouteille d'eau fermée et plongée dans une piscine : cette dernière constitue le réservoir. Il est clair que même si la bouteille est initialement à u

Procrustes Metrics and Optimal Transport for Covariance Operators

Valentina Masarotto

Covariance operators play a fundamental role in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on the space of such operators, as well as with their intrinsic infinite-dimensionality. We will show that we can identify the space of infinite-dimensional covariance operators equipped with the Procrustes size-and-shape metric from shape theory, with that of centred Gaussian processes, equipped with the Wasserstein metric of optimal transportation. We then describe key geometrical and topological aspects of the space of covariance operators endowed with the Procrustes metric. Through the notion of multicoupling of Gaussian measures, we establish existence, uniqueness and stability for the Fréchet mean of covariance operators with respect to the Procrustes metric. Furthermore, we will provide generative models that are canonical for such metric. We then turn to the problem of comparing several samples of stochastic processes with respect to their second-order structure, and we subsequently describe the main modes of variation in this second order structure. These two tasks are carried out via an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators respectively. In order to perform ANOVA, we introduce a novel approach based on optimal (multi)transport and identify each covariance with an optimal transport map. These maps are then contrasted with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, outperforms the state-of-the-art in the functional case. If the null hypothesis postulating equality of the operators is rejected, thanks to a geometric interpretation of the transport maps we can construct a PCA on the tangent space with the aim of understanding the sample variability. Finally, we provide a further example of use of the optimal transport framework, by applying it to the problem of clustering of operators. Two different clustering algorithms are presented, one of which is innovative. The transportation ANOVA, PCA and clustering are validated both on simulated scenarios and real dataset.

EPFL2019

Ab Initio Derived Force Fields for Predicting CO2 Adsorption and Accessibility of Metal Sites in the Metal-Organic Frameworks M-MOF-74 (M = Mn, Co, Ni, Cu)

Peng Bai, Wendy Lee Queen

Metal-organic frameworks (MOFs) are versatile nanoporous materials that have gained significant interest as low heat capacity, high selectivity sorbents for CO2 capture applications. Large-scale atomistic simulations for identifying high-performance MOFs are possible, but are limited to systems for which existing molecular mechanics force fields describe the interactions between the guest and framework atoms with sufficient accuracy. However, standard force fields are not applicable to cases involving coordinatively unsaturated metal centers that can strongly bind specific sorbate molecules. It has been previously shown that improved force fields can be derived from quantum mechanical calculations. In this work, we derived force fields for an isostructural series of MOFs, M-MOF-74, where M = Mn, Co, Ni, and Cu, from first principles. Monte Carlo calculations in the Gibbs ensemble were used to calculate the CO2 adsorption isotherms in order to assess the quality of the derived force field parameters and to determine a generally applicable procedure for obtaining a reliable force field for a targeted MOF and adsorbate system. The computed CO2 adsorption isotherms for the different M-MOF-74 members agree with experimental measurements at low loading and show that Ni-MOF-74 possesses the highest affinity toward CO2 and Cu-MOF-74 the weakest. In addition, we explored the source of open metal site and pore inaccessibility in these materials and quantified its impact on adsorption, especially the discrepancies often observed between experiments and simulations at high loadings.

Amer Chemical Soc2015

Feedback topology and XOR-dynamics in Boolean networks with varying input structure

Carlo Maffi

We analyze a model of fixed in-degree random Boolean networks in which the fraction of input-receiving nodes is controlled by the parameter gamma. We investigate analytically and numerically the dynamics of graphs under a parallel XOR updating scheme. This scheme is interesting because it is accessible analytically and its phenomenology is at the same time under control and as rich as the one of general Boolean networks. We give analytical formulas for the dynamics on general graphs, showing that with a XOR-type evolution rule, dynamic features are direct consequences of the topological feedback structure, in analogy with the role of relevant components in Kauffman networks. Considering graphs with fixed in-degree, we characterize analytically and numerically the feedback regions using graph decimation algorithms (Leaf Removal). With varying gamma, this graph ensemble shows a phase transition that separates a treelike graph region from one in which feedback components emerge. Networks near the transition point have feedback components made of disjoint loops, in which each node has exactly one incoming and one outgoing link. Using this fact, we provide analytical estimates of the maximum period starting from topological considerations.

2009

Procrustes Metrics and Optimal Transport for Covariance Operators

Valentina Masarotto

EPFL2019